The eta invariant and $$\tilde K$$ O of lens spaces

Gilkey, Peter Β.

doi:10.1007/bf01162239

Cited by 5 publications

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“…Let res : C(B) → C(S m−1 ) denote the restriction map. We then obtain a map It is shown in [2] (see also [26,Theorem 0.1]) that this map is surjective. Using the bundle structure of C(B) ⋊ G ∼ = C(B ⋊ G M n (C)), this map can be described as follows: First of all consider C(B ⋊ G M n (C)) as a bundle over [0, 1] with fibre C * (G) ∼ = M n (C) G at 0 and fibre C(S m−1 ) ⋊ G ∼ = C(S m−1 × G M n (C)) at every t = 0.…”

Section: P 132]) Its Inverse Indmentioning

confidence: 99%

“…Thus the existence of a trivialization together with surjectivity of the map res * in (5.23) would imply that K 0 (G\S m−1 ) ∼ = Z • 1 G\S m−1 . But one can see in the appendix of [26] that K 0 (G\S m−1 ) := K 0 (G\S m−1 )/Z • 1 G\S m−1 is non-trivial (of finite order) for many choices of groups Z n acting on a suitable R m . For instance: if m = 4 and n = 2, we get K 0 (G\S 3 ) ∼ = Z/2.…”

Section: P 132]) Its Inverse Indmentioning

confidence: 99%

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Structure and K-theory of crossed products by proper actions

Emerson¹,

Echterhoff²

2010

Preprint

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We study the C*-algebra crossed product C 0 (X) ⋊ G of a locally compact group G acting properly on a locally compact Hausdorff space X. Under some mild extra conditions, which are automatic if G is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular with respect to its fibring over the quotient space G\X. We also give some results on the K-theory of such C*-algebras. These more or less compute the K-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely K-theoretic proof of a result due to Paul Baum and Alain Connes on K-theory with complex coefficients of crossed products by finite groups.K * (C 0 (X)⋊G for finite group actions, is concentrated in the problem of computing the torsion subgroup. We do not shed much light on this problem.This paper is to some extent expository. Our goal is to provide a readable synopsis of what is known, and what can be proved without too much difficulty, about crossed products by proper actions, and their K-theory. The paper is organized as follows: after giving some preliminaries on proper actions in §1 we give a detailed discussion of the bundle structure of C 0 (X) ⋊ G in §2. In §3 we discuss the natural bijection between Prim(C 0 (X) ⋊ G) and the quotient space G\ Stab(X) and in §4 we show that this bijection is a homeomorphism for a quite naturally defined topology on Stab(X) if the action satisfies the slice property. All K-theoretic discussions can be found in §5.All spaces considered in this paper with the obvious exceptions of primitive ideal spaces of C*-algebras and the like, are assumed locally compact Hausdorff.

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Section: P 132]) Its Inverse Indmentioning

confidence: 99%

Section: P 132]) Its Inverse Indmentioning

confidence: 99%

Structure and K-theory of crossed products by proper actions

Emerson¹,

Echterhoff²

2010

Preprint

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The eta function and eta invariant of Z2r-manifolds

Podestá¹

2017

Differential Geometry and its Applications

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A. We compute the eta function η(s) and its corresponding η-invariant for the Atiyah-Patodi-Singer operator D acting on an orientable compact flat manifold of dimension n = 4h − 1, h ≥ 1, and holonomy group F ≃ Z2r , r ∈ N. We show that η(s) is a simple entire function times L(s, χ4), the L-function associated to the primitive Dirichlet character modulo 4. The η-invariant is 0 or equals ±2 k for some k ≥ 0 depending on r and n. Furthermore, we construct an infinite family F of orientable Z2r -manifolds with F ⊂ SO(n, Z). For the manifolds M ∈ F we have η(M ) = − 1 2 |T |, where T is the torsion subgroup of H1(M, Z), and that η(M ) determines the whole eta function η(s, M ). IEta series and η-invariant. Let M be an oriented compact Riemannian manifold of dimension n = 4h − 1, h ≥ 1, and consider the Atiyah-Patodi-Singer operator D (APS-operator for short) defined on the space of smooth even formswhere Ω 2p (M ) denotes the set of degree 2p forms. This operator is closely related to the signature operator. In fact, D is the tangential boundary operator of the signature operator S acting on a 4h-dimensional manifoldM having M as its boundary.By compactness of M , D has a discrete spectrum, Spec D (M ), of real eigenvalues λ with finite multiplicity d λ which accumulate only at infinity. The eta series (1.1) η(s) = 0 =λ∈Spec D (M ) sign(λ) |λ| −s , Re(s) > n, defines a holomorphic function having a meromorphic continuation to C, also denoted by η(s), having (possibly) simple poles in the set {n − k : k ∈ N 0 }. Remarkably, η(s) is holomorphic at s = 0 and the value η = η(0) is called the eta invariant of D.Both D and η(s) were first introduced and studied by Atiyah, Patodi and Singer in a sequel of 3 classical papers [1], where they also proved the regularity of η(s) at the origin in the case of odd dimension. The finiteness of η in any dimension is due to P. B. Gilkey ([9]). Actually, the results in [1] and [9] are valid for arbitrary elliptic differential operators.Eta series and η-invariants have been an active area of research since their appearance in [1]. A lot of progress have been made mainly by Peter Gilkey, Werner Müller, Xianzhe Dai, Weiping Zhang, Robert Meyerhoff, Mingquing Ouyang and Sebastian Goette among others. Eta series and η-invariants have been studied in several contexts. For instance, in equivariant settings (Donelly '76, Zhang '90 and Goette '99, '00, '09), in relation to connective K-theory (Gilkey '84 and Barrera-Yañez-Gilkey '99, '03), manifolds with boundary (Müller '93, '94, Bunke '95, '15 and Dai '02, '06) and cobordism Gilkey '88, '88, '97, '96 and Dai '05), flat vector bundles (Zhang '04 and Ma-Zhang '06, '06, '08), even dimensions (Gilkey

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Structure and K-theory of crossed products by proper actions

Echterhoff

Emerson

2011

Expositiones Mathematicae

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The eta invariant and $$\tilde K$$ O of lens spaces

Cited by 5 publications

References 7 publications

Structure and K-theory of crossed products by proper actions

Structure and K-theory of crossed products by proper actions

The eta function and eta invariant of Z2r-manifolds

Structure and K-theory of crossed products by proper actions

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